Question

Test the claim that the proportion of men who own cats is larger than 20% at the .005 significance level. The null and alternative hypothesis would be:_____.

A. H0: p = 0.2
Ha: p is not equal to 0.2.
B. H0: mu = 0.2
Ha: mu is not equal to 0.2.
C. H0: p = 0.2
Ha: p < 0.2.
D. H0: mu = 0.2
Ha: mu < 0.2
E. H0: mu = 0.2
Ha: mu > 0.2.
The test statistic is:____.
The critical value is:____.
Based on this we (to 2 decimals) (to 2 decimals):_____.
a. reject the null hypothesis.
b. fail to reject the null hypothesis.

Answer 1

`C. \\ \\ H_o: p = 0.2 \\ \\ Ha: p < 0.2`

Explanation:

From the given information:

Null & Alternative hypothesis is:

`\ \\ H_o: p = 0.2 \\ \\ Ha: p < 0.2`

Since the alternative hypothesis is less than 0.2;

Then, the test is left-tailed.

The level of significance ∝ = 0.005

Let assume that:

the sample size n of the people = 55 and there are 19 owned cats;

Then:

Test statistics:

`Z =\frac{\hat p - p}{\sqrt{(p(1-p))/(n)}}`

`Z =\frac{0.19 - 0.2}{\sqrt{(0.2(1-0.2))/(55)}}`

`Z =\frac{- 0.01}{\sqrt{(0.16)/(55)}}`

`Z =(- 0.01)/(√(0.002909))`

Z = - 0.18540

Z = - 0.19

At ∝ = 0.005; the critical value
`Z_(\alpha/2)= Z_(0.005/2)= -2.58`

Since the value of the test statistics is greater than the critical value; then, we fail to reject
`H_o` i.e the null hypothesis.